Introduction to number theory lecture 47. The prime number theorem

Introduction to number theory lecture 47. The prime number theorem

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematicsPBHNumber theory
🎙 Richard E Borcherds 👥 82K 📅 April 12, 2022 ⏱ 27 min 👁 10K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

prime number theoremRiemann zeta functionanalytic number theoryNewman's tauberian theoremSelberg's proof

Summary

This lecture, part of a Berkeley undergraduate number theory course, provides an overview and sketch of the proof of the prime number theorem. The theorem states that the number of primes less than x, denoted π(x), is asymptotic to x/log x. The lecturer begins by recalling elementary upper and lower bounds for π(x) that are within a constant factor of the true value. He then explains that the proof uses the Riemann zeta function, specifically its logarithmic derivative, and introduces the von Mangoldt function and the Chebyshev function ψ(x). The proof outline consists of five steps: (1) showing that ζ(s) has no zeros on the line Re(s)=1, (2) applying Newman’s tauberian theorem, (3) using this to show that an integral involving ψ(x) converges, (4) deducing that ψ(x) is asymptotic to x, and (5) concluding that π(x) is asymptotic to x/log x. The lecturer also discusses the history, including the original proofs by Hadamard and de la Vallée Poussin, and the later elementary proof by Selberg, along with a priority dispute with Erdős. He explains the convergence properties of Dirichlet series and the analytic continuation of ζ(s) to Re(s)>0. The lecture sets the stage for a more detailed proof in the next lecture.

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Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and insightful overview of the prime number theorem, emphasizing the key ideas and the structure of the proof. The argumentation is solid, as the lecturer carefully explains why each step is necessary and how they fit together. He highlights the central difficulty (showing no zeros on Re(s)=1) and the role of Newman’s tauberian theorem. The historical context and the discussion of Selberg’s elementary proof add value, giving a broader perspective. The presentation is well-organized, with a logical flow from background to proof sketch.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is scientifically rigorous, as expected from a university course by a leading expert. The lecturer references standard textbooks (Niven, Zuckerman, Montgomery) and a specific article by Zagier (linked in the description). The sources are appropriate and credible. The title accurately reflects the content, which is an introduction to the prime number theorem. The lecture does not include any promotional content.

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Title / Content Match

The title accurately reflects the content, which is an introduction to the prime number theorem.

Quality & Reliability

9/10

Lecture by a renowned mathematician, part of a university course, with references to standard literature and a peer-reviewed article. The content is rigorous and well-structured.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Concurring Sources

  • An Introduction to the Theory of Numbers — Textbook by Niven, Zuckerman, and Montgomery, referenced as the course textbook.

Contribution & Novelties

This lecture provides a clear and concise overview of the prime number theorem, emphasizing the key ideas and the structure of the proof. It is particularly valuable for students and enthusiasts who want to understand the theorem without delving into all technical details. The lecturer’s explanation of the role of the Riemann zeta function and Newman’s tauberian theorem is illuminating.

Pour aller plus loin :

  • Prime number theorem — Wikipedia article providing a comprehensive overview and history.
  • Riemann zeta function — Wikipedia article on the zeta function, including its properties and the Riemann hypothesis.
  • Newman’s tauberian theorem — Wikipedia article explaining the theorem and its applications.
  • Selberg’s identity — Wikipedia article on the identity used in Selberg’s elementary proof.

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Radar Profile

The radar profile shows high scores in all dimensions, with particularly strong quality of information and reliability. The lecture is technically advanced but accessible, and the content is well-supported by references.

Reliability 9/10